class 11 notes

Class 11 Physics Notes Chapter 1 (Chapter 1) – Physics Part-II Book

Philoid

Philoid

— 8 min read

Physics Part-II
Alright class, let's get started with the first chapter from your Physics Part-II textbook, which is Chapter 9: Mechanical Properties of Solids. This is a crucial chapter, not just for your Class 11 exams, but also forms the basis for many concepts asked in various government examinations like NEET, JEE (though less direct), SSC, Railways, and other state-level exams where Physics is a component. Pay close attention to the definitions, formulas, units, and the stress-strain curve.

Chapter 9: Mechanical Properties of Solids - Detailed Notes

1. Introduction

  • Solid: A state of matter characterized by a definite shape and volume. The constituent atoms or molecules are held in fixed positions by strong interatomic or intermolecular forces and oscillate about their mean positions.
  • Deforming Force: An external force applied to a body which changes its size, shape, or both.
  • Elasticity: The property of a material body by virtue of which it regains its original shape and size after the removal of the deforming force. Examples: Steel, rubber (to a certain limit), quartz fibre.
  • Plasticity: The property of a material body by virtue of which it does not regain its original shape and size even after the removal of the deforming force. It undergoes permanent deformation. Examples: Putty, mud, plasticine.
  • Perfectly Elastic Body: A body that completely regains its original configuration immediately after the removal of the deforming force. (Ideal concept, quartz fibre is very close).
  • Perfectly Plastic Body: A body that does not show any tendency to regain its original configuration after the removal of the deforming force. (Ideal concept, putty is close).

2. Elastic Behaviour of Solids

  • In a solid, atoms/molecules are bound by interatomic/intermolecular forces.
  • These forces can be visualized as springs connecting adjacent atoms.
  • When a deforming force is applied, these atoms are displaced, stretching or compressing these 'springs'.
  • Internal restoring forces develop within the body, opposing the deforming force.
  • If the deforming force is removed, these restoring forces bring the atoms back to their equilibrium positions, and the body regains its original shape/size (Elasticity).
  • If the deforming force is too large, the atoms are displaced permanently (Plasticity).

3. Stress

  • Definition: The internal restoring force developed per unit area of cross-section of a deformed body.
  • In equilibrium, the magnitude of the internal restoring force is equal to the magnitude of the applied deforming force.
  • Formula: Stress (σ) = Restoring Force (F) / Area (A)
  • Unit: N/m² or Pascal (Pa) (SI); dyne/cm² (CGS)
  • Dimension: [ML⁻¹T⁻²]
  • Types of Stress:
    • Longitudinal Stress (or Normal Stress): When the deforming force acts perpendicular (normal) to the surface area.
      • Tensile Stress: If the force causes an increase in length.
      • Compressive Stress: If the force causes a decrease in length.
    • Shearing Stress (or Tangential Stress): When the deforming force acts parallel (tangential) to the surface area, causing a change in shape but not volume.
    • Hydraulic Stress (or Volume Stress): When forces act uniformly and normally all over the surface of the body, causing a change in volume but not shape (e.g., a body immersed in a fluid under pressure). It's numerically equal to the pressure (P).

4. Strain

  • Definition: The ratio of the change in configuration (dimension like length, shape, or volume) to the original configuration of the body.
  • Formula: Strain (ε) = Change in Dimension / Original Dimension
  • Unit: Dimensionless (it's a ratio).
  • Types of Strain:
    • Longitudinal Strain: Ratio of change in length (ΔL) to the original length (L).
      • ε<0xE1><0xB5><0xAB> = ΔL / L
    • Shearing Strain: The angle (θ) through which a line originally perpendicular to the fixed surface is turned due to tangential stress. For small angles, tan θ ≈ θ.
      • ε<0xE2><0x82><0x9B> = tan θ ≈ θ = Δx / L (where Δx is the relative displacement of the face on which force is applied and L is the distance between the faces).
    • Volumetric Strain: Ratio of change in volume (ΔV) to the original volume (V).
      • ε<0xE1><0xB5><0xA3> = ΔV / V (A decrease in volume leads to negative strain).

5. Hooke's Law

  • Statement: For small deformations, stress is directly proportional to strain.
  • Formula: Stress ∝ Strain or Stress = E × Strain
  • E: Modulus of Elasticity (or Coefficient of Elasticity). Its value depends on the nature of the material and the type of stress/strain.
  • Unit of E: Same as stress (N/m² or Pa).
  • Dimension of E: [ML⁻¹T⁻²].
  • Validity: This law is valid only within the 'limit of proportionality' (often close to the 'elastic limit').

6. Stress-Strain Curve

  • A graph plotted between stress (usually on y-axis) and strain (usually on x-axis) for a material under load (typically tensile stress).
  • Key Points on the Curve (for a ductile material like mild steel):
    • O to A (Proportional Limit): Stress is directly proportional to strain (Hooke's Law obeyed). The graph is a straight line.
    • A to B (Elastic Limit / Yield Point - approx): Stress is not proportional to strain, but the material returns to its original state when the load is removed. Point B is the Elastic Limit or Yield Point. The stress corresponding to the yield point is called Yield Strength (Sy).
    • B to D (Plastic Region): Beyond the elastic limit, the material undergoes permanent deformation. Even if the load is removed, it won't regain its original shape/size. Strain increases rapidly for small increases in stress.
    • Point D (Ultimate Tensile Strength): The maximum stress the material can withstand before starting to neck down (local thinning). (Stress = Su)
    • Point E (Fracture Point): The point where the material breaks.
  • Ductile Materials: Materials that show large plastic deformation before fracture (e.g., copper, aluminum, mild steel). They can be drawn into wires. Large region between B and E.
  • Brittle Materials: Materials that fracture soon after the elastic limit is crossed (e.g., glass, ceramics, cast iron). Small or negligible region between B and E.
  • Elastomers: Materials (like rubber) that can be stretched to cause large strains but still return to original shape. They do not obey Hooke's law over most regions, and the stress-strain curve is not linear.

7. Moduli of Elasticity

  • a) Young's Modulus (Y):
    • Ratio of longitudinal stress to longitudinal strain within the elastic limit.
    • Measures resistance to change in length.
    • Y = Longitudinal Stress / Longitudinal Strain = (F/A) / (ΔL/L) = FL / AΔL
    • Higher Y means more rigid (less elastic in common terms, but more elastic in physics terms as it resists deformation better). Steel has a higher Y than copper or aluminum.
  • b) Shear Modulus (G) or Modulus of Rigidity:
    • Ratio of shearing stress to shearing strain within the elastic limit.
    • Measures resistance to change in shape.
    • G = Shearing Stress / Shearing Strain = (F<0xE1><0xB5><0xA2><0xE1><0xB5><0x8A><0xE1><0xB5><0x8A> / A) / θ ≈ (F<0xE1><0xB5><0xA2><0xE1><0xB5><0x8A><0xE1><0xB5><0x8A> / A) / (Δx/L)
    • Liquids and gases have G = 0 (they cannot sustain shear stress).
  • c) Bulk Modulus (B):
    • Ratio of hydraulic (volume) stress to the corresponding volumetric strain within the elastic limit.
    • Measures resistance to change in volume.
    • B = Hydraulic Stress / Volumetric Strain = -ΔP / (ΔV/V)
    • The negative sign indicates that an increase in pressure (ΔP > 0) causes a decrease in volume (ΔV < 0), making B positive.
    • Higher B means less compressible. Solids > Liquids > Gases in terms of B.
    • Compressibility (k): The reciprocal of Bulk Modulus. k = 1/B. Measures how easily a substance can be compressed.

8. Poisson's Ratio (σ or ν)

  • When a wire is stretched (longitudinal strain), its diameter decreases (lateral strain).
  • Definition: Within the elastic limit, the ratio of lateral strain to the longitudinal strain.
  • Formula: σ = Lateral Strain / Longitudinal Strain = -(ΔD/D) / (ΔL/L)
    • ΔD/D is the lateral strain (change in diameter / original diameter).
    • ΔL/L is the longitudinal strain (change in length / original length).
    • The negative sign ensures σ is positive (as ΔD is negative when ΔL is positive).
  • Unit: Dimensionless.
  • Theoretical Range: -1 < σ < 0.5
  • Practical Range (for most materials): 0 < σ < 0.5 (Typically between 0.2 and 0.4).

9. Elastic Potential Energy in a Stretched Wire

  • Work has to be done by the applied force to stretch a wire, which gets stored as elastic potential energy (U) in the wire.
  • Work Done (W) = Average Force × Elongation = (0 + F)/2 × ΔL = 1/2 × F × ΔL
  • Potential Energy (U): U = 1/2 × F × ΔL
  • Using Stress = F/A and Strain = ΔL/L, we can write:
    • U = 1/2 × (Stress × A) × (Strain × L)
    • U = 1/2 × Stress × Strain × (A × L)
    • U = 1/2 × Stress × Strain × Volume
  • Elastic Potential Energy Density (u): Potential energy stored per unit volume.
    • u = U / Volume = 1/2 × Stress × Strain
    • Also, u = 1/2 × (Y × Strain) × Strain = 1/2 × Y × (Strain)²
    • Also, u = 1/2 × Stress × (Stress / Y) = 1/2 × (Stress)² / Y

10. Applications of Elastic Behaviour

  • Design of Bridges and Buildings: Beams and columns are designed to withstand loads without excessive bending or buckling. Knowledge of Young's modulus and yield strength is crucial. I-shaped girders are used to provide strength with less material (reduces weight and cost) by maximizing the distance of material from the neutral axis.
  • Cranes: The thickness of the metallic rope used in cranes is determined based on the elastic limit of the material and the maximum load it needs to lift, ensuring a safety factor.
  • Maximum Height of a Mountain: Limited by the elastic properties (specifically, the yield strength against compression) of the rock at the base.

Multiple Choice Questions (MCQs) for Practice

  1. The property of a material by virtue of which it regains its original shape and size after the removal of deforming force is called:
    a) Plasticity
    b) Elasticity
    c) Ductility
    d) Malleability

  2. The SI unit of stress is the same as that of:
    a) Force
    b) Pressure
    c) Strain
    d) Young's Modulus

  3. According to Hooke's law of elasticity, if stress is increased, the ratio of stress to strain:
    a) Increases
    b) Decreases
    c) Becomes zero
    d) Remains constant

  4. Which of the following has the highest Young's Modulus?
    a) Rubber
    b) Copper
    c) Steel
    d) Wood

  5. Shearing strain is given by: (where symbols have their usual meaning)
    a) ΔL / L
    b) ΔV / V
    c) F / A
    d) Δx / L (or tan θ)

  6. The Bulk modulus for an incompressible liquid is:
    a) Zero
    b) Unity
    c) Infinite
    d) Between 0 and 1

  7. Poisson's ratio (σ) is defined as the ratio of:
    a) Longitudinal strain to lateral strain
    b) Lateral strain to longitudinal strain
    c) Shearing stress to shearing strain
    d) Longitudinal stress to longitudinal strain

  8. In the stress-strain curve for a metal wire, the point where the material breaks is called the:
    a) Elastic limit
    b) Yield point
    c) Ultimate tensile strength point
    d) Fracture point

  9. The energy stored per unit volume in a stretched wire is:
    a) 1/2 × Stress × Strain
    b) Stress × Strain
    c) 1/2 × Y × (Stress)²
    d) 1/2 × Load × Extension

  10. A wire of length L and radius r is fixed at one end and a force F is applied to the other end, producing an extension ΔL. Which of the following graphs is a straight line?
    a) F vs L
    b) F vs ΔL
    c) F vs 1/ΔL
    d) F vs r

Answers to MCQs:

  1. b) Elasticity
  2. b) Pressure (Both are Force/Area, N/m² or Pa) (d) Young's Modulus also has the same unit. However, Pressure is a more fundamental quantity with the same unit. In exams, both might be considered correct, but Pressure is a direct match based on definition F/A. Let's stick with Pressure as the primary answer. Self-correction: Young's Modulus is Stress/Strain, and Strain is dimensionless, so Y has the same unit as Stress. Both b and d have the same unit. Usually, questions like this relate stress directly to pressure. Let's re-evaluate. Stress is internal restoring force per area. Pressure is external force per area. Dimensionally and unit-wise they are identical. Young's Modulus is also dimensionally identical. Given the options, both Pressure and Young's Modulus are valid unit-wise. However, Pressure is conceptually closer to force per area. Let's choose (b) but acknowledge (d) is also correct unit-wise. Final decision: Go with (b) Pressure, as it's a more direct comparison of F/A.
  3. d) Remains constant (This ratio is the modulus of elasticity, which is constant within the proportional limit).
  4. c) Steel
  5. d) Δx / L (or tan θ)
  6. c) Infinite (Incompressible means ΔV = 0 for any finite pressure change ΔP. B = -ΔP / (ΔV/V), so if ΔV=0, B tends to infinity).
  7. b) Lateral strain to longitudinal strain
  8. d) Fracture point
  9. a) 1/2 × Stress × Strain
  10. b) F vs ΔL (According to Hooke's Law, F/A = Y (ΔL/L), so F = (AY/L) * ΔL. Since A, Y, L are constant for a given wire, F is directly proportional to ΔL within the elastic limit).

Remember to revise these concepts thoroughly. Focus on understanding the definitions and the physical meaning behind the formulas. Good luck with your preparation!

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